In mobile radio communication systems that use Multi-User MIMO, MU-MIMO, several users are scheduled to transmit simultaneously during the same time and frequency interval. These users can be distinguished by allocating different, and preferably orthogonal, reference signals to them such that all channels from all transmit antennas to all receive antennas can be estimated. The reference signals, or training sequence, are sequences known to the receiver and they are transmitted together with user signals such that the receiver can estimate distortions of the received signal after it is transmitted.
MU-MIMO forms part of, e.g., the third generation partnership project, 3GPP, standards including the global system for mobile communication, GSM, as well as the long term evolution, LTE, standard.
A radio communication between a UE and a base-station will be affected by multi path propagation, fading, frequency errors, round trip times etc. This communication channel is often referred to as an air interface, and causes bit and block errors on information transmitted. A receiver is designed in order to reduce bit error and block error rates, which among others, includes channel estimation, demodulation and decoding blocks. Other processing blocks commonly used in uplink receivers includes spatial and temporal whitening and pre-filtering, as illustrated in FIG. 3.
In order to make it possible to reduce such bit error and block error rates for data bursts in the communication between a UE and a base-station, so-called training sequences are typically inserted into the data bursts. The training sequence is a sequence of information that is known to the receiver. This is inserted into a transmission slot together with user data such that the receiver can estimate distortions of the received signal after it has been transmitted. For example, the training sequence can be inserted in the middle of one burst with user data both prior to and subsequent to the training sequence. When equalizing the user data, the user data signals prior to and subsequent to the training sequence are usually equalized with Viterbi equalizers, one for the “prior to” data part and one for the “subsequent to” data part. The “prior to” data part is in the following description referred to as being associated with a “backward” direction and the “subsequent to” data part is in the following description referred to as being associated with a “forward” direction.
The received signal in a MIMO system with N transmitters and M receiver antennas subject to an ISI channel can be described as
                              Y          ⁡                      (            n            )                          =                                                            ∑                                  k                  =                  0                                L                            ⁢                                                          ⁢                                                H                  ⁡                                      (                    k                    )                                                  ⁢                                  S                  ⁡                                      (                                          n                      -                      k                                        )                                                                        +                          W              ⁡                              (                n                )                                              =                                                    H                ⁡                                  (                                      z                                          -                      1                                                        )                                            ⁢                              S                ⁡                                  (                  n                  )                                                      +                          W              ⁡                              (                n                )                                                                        (        1        )                where W(n) is a M×1 vector with additive noise and interference, S(n) is a N×1 vector with transmitted signalsS(n)=[s1(n)s2(n) . . . sN(n)]T  (2)and H(z−1) is an M×N channel matrix with polynomial entries in the unity delay operator z1, i.e. z−1s(n)=s(n−1), such that the element in row i and column j equalshi,j(z−1)=hi,j,0+hi,j,1z−1+hi,j,2z−2+ . . . +hi,j,Lz−L  (3)
A polynomial in the unity delay operator is thus a compact description of a time invariant FIR filter.
Polynomial QR decomposition is described in the prior art:    J. Foster, J. McWhriter and J. Chambers, “A polynomial matrix QR decomposition with application to MIMO channel equalization”, Forty-First Asilomar Conference on Signals, Systems and Computers, (ACSSC 2007), Pacific Grove, Calif., November 4-7, pp. 1379-1383,    M. Davies, S. Lambotharan, and J. Foster, “A polynomial QR decomposition based turbo equalization technique for frequency selective MIMO channels”, Proceedings of 2009 69th Vehicular Technology Conference (VTC 2009), Barcelona, Spain, 2009, and    D. Cescato and H. Bölcskei, “QR decomposition of Laurent polynomial matrices sampled on the unit circle”, presented at IEEE Transactions on Information Theory, 2010, pp. 4754-4761.
Using polynomial QR decomposition, as described in the prior art, of the channel matrix results inQ(z−1)R(z−1)=H(z−1)  (4)    where R(z−1) is an upper triangular polynomial matrix of size N×N
                              R          ⁡                      (                          z                              -                1                                      )                          =                  [                                                                                          r                                          1                      ,                      1                                                        ⁡                                      (                                          z                                              -                        1                                                              )                                                                                                                    r                                          1                      ,                      2                                                        ⁡                                      (                                          z                                              -                        1                                                              )                                                                              …                                                                                  r                                          1                      ,                      N                                                        ⁡                                      (                                          z                                              -                        1                                                              )                                                                                                      0                                                                                  r                                          2                      ,                      2                                                        ⁡                                      (                                          z                                              -                        1                                                              )                                                                                                                                                            ⋮                                                                    ⋮                                                                                                                          ⋱                                                                                                                                                  0                                            …                                            0                                                                                  r                                          N                      ,                      N                                                        ⁡                                      (                                          z                                              -                        1                                                              )                                                                                ]                                    (        5        )                and Q(z−1) is a paraunitary polynomial matrix of size M×N. A matrix Q(z−1) is a paraunitary polynomial matrix ifQ(z−1)Q*T(z)=IM,Q*T(z)Q(z−1)=IN  (6)    where Q*T and IN are unitary matrices of size M and N, respectively. The paraconjugate of a matrix is defined as{tilde over (Q)}(z−1)Q*T(z).  (7)    where Q*T (z) denotes a matrix with conjugate of the coefficients, z−1 replaced by z and transpose of the matrix Q(z−1). From (4) and (6) follows that a multiplication of (1) with {tilde over (Q)}(z−1) results inYP(n)={tilde over (Q)}(z−1)Y(n)=R(z−1)S(n)+{tilde over (Q)}(z−1)W(n)  (8)i.e. a signal with upper triangular channel matrix R(z1). The multiplication with the polynomial matrix {tilde over (Q)}(z−1) is referred to as a MIMO pre-filter.
Assume that the additive noise and interference W(n) is temporally white and uncorrelated between the antenna branches i.e.E{W(n)W*T(n−m)}=IM·δ(m)  (9)
This can e.g. be accomplished by spatial-temporal whitening prior to (1), as illustrated by the whitening function 302 in FIG. 3. Since {tilde over (Q)}(z−1) is paraunitary, the whiteness and uncorrelateness of the additive nose and interference in (8) is preserved i.e.E{{tilde over (Q)}(z−1)W(n)({tilde over (Q)}(z)W(n−m)*T)}={tilde over (Q)}(z−1){tilde over (Q)}(z)·(m)=IN·δ(m)  (10)
A turbo equalizer can now be used for equalization and demodulation of the transmitted symbols in S(n), e.g. as outlined in Davies et al.
A polynomial decomposition is proposed in Foster et al. which is based on Givens rotations. Here several iterations are needed in order for the QR decomposition to converge.
A polynomial QR-decomposition is also described in Cescato and Bölcskei. Here, the channel H(ωn)=H(z1)|z=ejwn is available for a number of discrete frequencies ωn. A QR-decomposition of these channel matrices is described such that an interpolation to any frequency is available.
The complexity of a demodulator typically increases with the length of the radio channel. In order to reduce computational complexity, a short channel model is beneficial. With a minimum phase channel model, most of the energy is concentrated in the first taps of the channel. However a polynomial QR decomposition as done in Foster et al. is based upon Givens rotations which do not guarantee minimum phase channel after pre-filtering.